3.5.32 \(\int \frac {(c+d x^2)^3}{x^{13/2} (a+b x^2)} \, dx\)

Optimal. Leaf size=305 \[ \frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}-\frac {2 c^3}{11 a x^{11/2}} \]

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Rubi [A]  time = 0.27, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {466, 461, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {2 c^3}{11 a x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]

[Out]

(-2*c^3)/(11*a*x^(11/2)) + (2*c^2*(b*c - 3*a*d))/(7*a^2*x^(7/2)) - (2*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*
a^3*x^(3/2)) + ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(15/4)*b^(1/4)) - ((b*
c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(15/4)*b^(1/4)) + ((b*c - a*d)^3*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(15/4)*b^(1/4)) - ((b*c - a*d)^3*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(15/4)*b^(1/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^{12} \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {c^3}{a x^{12}}+\frac {c^2 (-b c+3 a d)}{a^2 x^8}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^4}+\frac {(-b c+a d)^3}{a^3 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {\left (2 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{7/2}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{7/2}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{7/2} \sqrt {b}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{7/2} \sqrt {b}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [C]  time = 0.38, size = 102, normalized size = 0.33 \begin {gather*} -\frac {2 \left (a c \left (3 a^2 \left (7 c^2+33 c d x^2+77 d^2 x^4\right )-33 a b c x^2 \left (c+7 d x^2\right )+77 b^2 c^2 x^4\right )+231 x^6 (b c-a d)^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{231 a^4 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]

[Out]

(-2*(a*c*(77*b^2*c^2*x^4 - 33*a*b*c*x^2*(c + 7*d*x^2) + 3*a^2*(7*c^2 + 33*c*d*x^2 + 77*d^2*x^4)) + 231*(b*c -
a*d)^3*x^6*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(231*a^4*x^(11/2))

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IntegrateAlgebraic [A]  time = 0.25, size = 206, normalized size = 0.68 \begin {gather*} -\frac {(a d-b c)^3 \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(a d-b c)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {2 c \left (21 a^2 c^2+99 a^2 c d x^2+231 a^2 d^2 x^4-33 a b c^2 x^2-231 a b c d x^4+77 b^2 c^2 x^4\right )}{231 a^3 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]

[Out]

(-2*c*(21*a^2*c^2 - 33*a*b*c^2*x^2 + 99*a^2*c*d*x^2 + 77*b^2*c^2*x^4 - 231*a*b*c*d*x^4 + 231*a^2*d^2*x^4))/(23
1*a^3*x^(11/2)) - ((-(b*c) + a*d)^3*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]
])/(Sqrt[2]*a^(15/4)*b^(1/4)) + ((-(b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]
*x)])/(Sqrt[2]*a^(15/4)*b^(1/4))

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fricas [B]  time = 1.49, size = 1866, normalized size = 6.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

1/462*(924*a^3*x^6*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*
c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*
c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4)*arctan((sqrt(a^8*sqrt(-(b^12*c^
12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5
+ 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^1
0 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b)) + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d
^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*x)*a^11*b*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^1
0*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^
5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15
*b))^(3/4) + (a^11*b^4*c^3 - 3*a^12*b^3*c^2*d + 3*a^13*b^2*c*d^2 - a^14*b*d^3)*sqrt(x)*(-(b^12*c^12 - 12*a*b^1
1*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^
6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b
*c*d^11 + a^12*d^12)/(a^15*b))^(3/4))/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d
^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d
^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)) + 231*a^3*x^6*(-(b^12*c^12 -
12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 92
4*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 1
2*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4)*log(a^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 -
220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 +
495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/
4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 231*a^3*x^6*(-(b^12*c^12 - 12*a*b^11*c^11*
d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d
^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11
+ a^12*d^12)/(a^15*b))^(1/4)*log(-a^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^
9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^
4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4) - (b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 4*(21*a^2*c^3 + 77*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)
*x^4 - 33*(a*b*c^3 - 3*a^2*c^2*d)*x^2)*sqrt(x))/(a^3*x^6)

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giac [B]  time = 0.51, size = 483, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} - \frac {2 \, {\left (77 \, b^{2} c^{3} x^{4} - 231 \, a b c^{2} d x^{4} + 231 \, a^{2} c d^{2} x^{4} - 33 \, a b c^{3} x^{2} + 99 \, a^{2} c^{2} d x^{2} + 21 \, a^{2} c^{3}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4
)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) - 1/2*sqrt(2)*((a*b^3)^(1
/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*s
qrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)
^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(a^4*b) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*
b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) - 2/231*(77*b^2*c^3
*x^4 - 231*a*b*c^2*d*x^4 + 231*a^2*c*d^2*x^4 - 33*a*b*c^3*x^2 + 99*a^2*c^2*d*x^2 + 21*a^2*c^3)/(a^3*x^(11/2))

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maple [B]  time = 0.02, size = 659, normalized size = 2.16 \begin {gather*} \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2}}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2}}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \,d^{2} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{4}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{4}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} c^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 a^{4}}-\frac {2 c \,d^{2}}{a \,x^{\frac {3}{2}}}+\frac {2 b \,c^{2} d}{a^{2} x^{\frac {3}{2}}}-\frac {2 b^{2} c^{3}}{3 a^{3} x^{\frac {3}{2}}}-\frac {6 c^{2} d}{7 a \,x^{\frac {7}{2}}}+\frac {2 b \,c^{3}}{7 a^{2} x^{\frac {7}{2}}}-\frac {2 c^{3}}{11 a \,x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x)

[Out]

1/2/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3-3/2/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)-1)*b*c*d^2+3/2/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b^2*c^2*d-1/
2/a^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b^3*c^3+1/4/a*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^
(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*d^3-3/4/a^2*(a/b)^(1/4)*2^(1/2
)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*b*c*d^2+3/4/a^3*
(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))
)*b^2*c^2*d-1/4/a^4*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*
x^(1/2)+(a/b)^(1/2)))*b^3*c^3+1/2/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3-3/2/a^2*(a/b
)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b*c*d^2+3/2/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)
^(1/4)*x^(1/2)+1)*b^2*c^2*d-1/2/a^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b^3*c^3-2/11*c^3
/a/x^(11/2)-2*c/a/x^(3/2)*d^2+2*c^2/a^2/x^(3/2)*b*d-2/3*c^3/a^3/x^(3/2)*b^2-6/7*c^2/a/x^(7/2)*d+2/7*c^3/a^2/x^
(7/2)*b

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maxima [A]  time = 2.37, size = 389, normalized size = 1.28 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a^{3}} - \frac {2 \, {\left (21 \, a^{2} c^{3} + 77 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 33 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4
) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)
*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(s
qrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3
*a^2*b*c*d^2 - a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a^3 - 2
/231*(21*a^2*c^3 + 77*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)*x^4 - 33*(a*b*c^3 - 3*a^2*c^2*d)*x^2)/(a^3*x^(11/2
))

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mupad [B]  time = 0.51, size = 1580, normalized size = 5.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x)

[Out]

- ((2*c^3)/(11*a) + (2*c^2*x^2*(3*a*d - b*c))/(7*a^2) + (2*c*x^4*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*a^3))/x
^(11/2) - (atan(((((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^
11*b^7*c^4*d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b
^3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^(1/
4)) + (((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*
d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48
*a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^(1/4)))/((((x^
(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a
^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c
^2*d + 48*a^15*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4)) - (((x^(1/2)*(16*a^9*
b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a^12*b^6*c^3*d^
3 + 240*a^13*b^5*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15
*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4))))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^
(1/4)) - (atan(((((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^1
1*b^7*c^4*d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^
3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2)*1i)/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4
)) + (((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d
^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*
a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2)*1i)/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4)))/((((x^(
1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a^
12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c^
2*d + 48*a^15*b^4*c*d^2)*1i)/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^(1/4)) - (((x^(1/2)*(16
*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a^12*b^6*c
^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c^2*d + 48
*a^15*b^4*c*d^2)*1i)/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^(1/4))))*(a*d - b*c)^3)/((-a)^(
15/4)*b^(1/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3/x**(13/2)/(b*x**2+a),x)

[Out]

Timed out

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